A NOTE ON A MULTIVALUED ITERATIVE EQUATION
LIN LI
Abstract. In this note, we consider a second order multivalued iterative equation, and the result on
decreasing solutions is given.
1. Introduction
Let X be a topological space and for integer n ≥ 0 the n-th iterate of a mapping f is defined
by f n = f ◦ f n−1 and f 0 = id, where ◦ denotes the composition of mappings and id denotes
the identity mapping. As an important class of functional equations [1, 2], the polynomial-like
iterative equation is a linear combination of iterates, which is of the general form
(1)
λ1 f (x) + λ2 f 2 (x) + . . . + λn f n (x) = F (x),
x ∈ X,
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where X is a Banach space or its closed subset, where F is a given mapping, f is an unknown
mapping, and λi (i = 1, . . . , n) are real constants. Equation (1) has been studied extensively on
the existence, uniqueness and stability of its solutions [1, 2], it was also considered in the class of
multifunctions [3].
Let I = [a, b] be a given interval and cc(I) denote the family of all nonempty convex compact
subsets of I. This family endowed with the Hausdorff distance is defined by
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h(A, B) = max {sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}},
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Received November 30, 2007; revised March 12, 2008.
2000 Mathematics Subject Classification. Primary 39B12, 37E05, 54C60.
Key words and phrases. iteration; functional equation; multifunctions; upper semicontinuity.
where d(a, B) = inf{|a − b| : b ∈ B}.
A multifunction F : I → cc(I) is decreasing (resp. strictly decreasing) if max F (x) ≤ min F (y)
(resp. max F (x) < min F (y)) for every x, y ∈ I with x > y.
Let Γ(I) be the family of all multifunctions F : I → cc(I) and Φ(I) be defined by
Φ(I) = {F ∈ Γ(I) : is USC, increasing, F (a) = {a}, F (b) = {b}},
and endowed with the metric
D(F1 , F2 ) = sup{h(F1 (x), F2 (x)) : x ∈ I},
∀F1 , F2 ∈ Φ(I).
In [3], the authors investigated the second order multivalued iterative equation
(2)
λ1 F (x) + λ2 F 2 (x) = G(x),
in an interval I = [a, b], and the following results were obtained:
Lemma 1. ([3, Lemma 1.]) The metric space (Φ(I), D) is complete.
Lemma 2. ([3, Lemma 2.]) If F, G ∈ Φ and F (x) ⊂ G(x) for all x ∈ I, then F = G.
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Lemma 3. ([3, Theorem 1.]) Let G ∈ Φ(I), λ1 > λ2 ≥ 0 and λ1 + λ2 = 1. Then the equation
(2) has a unique solution F ∈ Φ(I).
In this paper, as in [3], we are still interested in a multivalued solution of the equation (2), and
the decreasing solutions are given.
2. Main Result
Let I = [−a, a] be a given interval and Ψ(I) be defined by
Ψ(I) = {F : I → cc(I), is USC, decreasing, F (−a) = {a}, F (a) = {−a}}.
We endow Ψ(I) with the metric
D(F1 , F2 ) = sup{h(F1 (x), F (x2 )) : x ∈ I},
∀F1 , F2 ∈ Ψ(I)
Obviously, by Lemma 1 the metric space (Ψ(I), D) is complete.
Lemma 4. If G, F ∈ Ψ(I) and F (x) ⊂ G(x) for all x ∈ I, then F = G.
This Lemma follows from Lemma 2. More concretely, we should consider the monotonicity of
decreasing.
Theorem 1. Let G ∈ Ψ(I), λ1 > 1/2, λ2 < 0 and λ1 − λ2 = 1. Then the equation (2) has a
unique solution F ∈ Ψ(I).
Proof. Define the mapping L : Ψ(I) → Γ(I)
LF (x) = λ1 x + λ2 F (x),
∀x ∈ I,
where F ∈ Ψ(I). Obviously, LF is USC and LF (−a) = −λ1 a + λ2 a = {−a}, LF (a) = λ1 a − λ2 a =
{a}. Moreover, for any x2 > x1 in I, we have max F (x2 ) − min F (x1 ) ≤ 0 since F is decreasing.
Therefore,
min LF (x2 ) − max LF (x1 ) = λ1 (x2 − x1 ) + λ2 (min F (x2 ) − max F (x1 ))
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≥ λ1 (x2 − x1 ) > 0
for λ1 > 0 and λ2 < 0, which implies that LF is strictly increasing and the multifunction (LF )−1
defined by (LF )−1 (y) = {x ∈ I : y ∈ LF (x)} is single-valued and continuous.
Define the mapping Υ : Ψ(I) → Γ(I) as
ΥF (x) = (LF )−1 (G(x)),
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−1
∀F ∈ Ψ(I),
∀x ∈ I,
Hence, ΥF is also USC and ΥF (−a) = (LF ) (G(−a)) = {a}, ΥF (a) = (LF )−1 (G(a)) = {−a}.
Moreover, ΥF is decreasing since (LF )−1 is increasing and G is decreasing.
Finally, by (c.f. [3, pp. 431–432]), we have
D(ΥF1 , ΥF2 ) = sup h((LF1 )−1 (G(x)), (LF2 )−1 (G(x)))
x∈I
≤
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1
sup h(LF1 (x), LF2 (x)),
λ1 x∈I
for every x ∈ I and F1 , F2 ∈ Ψ(I). Hence, we obtain that
1
sup h(LF1 (x), LF2 (x))
D(ΥF1 , ΥF2 ) ≤
λ1 x∈I
1
≤
sup h(λ1 x + λ2 F1 (x), λ1 x + λ2 F2 (x))
λ1 x∈I
|λ2 |
=
sup h(F1 (x), F2 (x))
λ1 x∈I
|λ2 |
≤
D(F1 , F2 )
λ
1
1
=
− 1 D(F1 , F2 )
λ1
< D(F1 , F2 )
which implies that Υ is a contraction. Therefore, by the Banach fixed point principle, Υ has a
unique fixed point F in Ψ(I), i.e.
(LF )−1 (G(x)) = F (x),
∀x ∈ I.
Consequently, by Lemma 4, we have
λ1 F (x) + λ2 F 2 (x) = G(x),
∀x ∈ I.
The proof is completed.
Acknowledgement. The author is very grateful to Professor Bing Xu (Sichuan University)
and the referee who checked her paper carefully and gave detailed suggestions.
1. Braon K., Jarczyk W., Recent results on functional equations in a single variable, perspectives and open problems,
Aequationes Math. 61 (2001), 1–48.
2. Kuczma M., Functional equations in a single variable, Monografie Mat. 46, PWN, Warszawa, 1968.
3. Nikodem K. and Weinian Zhang, On a multivalued iterative equation, Publ. Math. Debrecen 64 (2004), 427–435.
Lin Li, Department of Mathematics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, P.
R. China
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